Given the joint pdf of $X_1$ and $X_2$ as $f(x_1,x_2)=\frac{1}{\pi}I_{(x_1^2+x_2^2<1)}$, find the joint pdf of $Y_1=\sqrt{X_1^2+X_2^2}$ and $Y_2=\frac{X_1}{\sqrt{X_1^2+X_2^2}}$.
Here is my attempt.
I transformed $X_1$ and $X_2$ into polar coordinates as $X_1=R \cos{\Theta}$ and $X_2=R \sin{\Theta}$, obtaining the joint pdf of $R$ and $\Theta$ as $pdf(r,\theta)=\frac{r}{\pi}I_{(0\leq r<1)}I_{(0\leq \theta<2\pi)}$.
Now I will define a function from $\{(r,\theta)^T:0\leq r <1, 0\leq \theta <2\pi\}$ to $\{(y_1,y_2)^T:0\leq y_1 <1, -1 \leq y_2 < 1\}$ as
$ \begin{cases} y_1=r\\ y_2=\cos \theta \end{cases} $, $ \begin{cases} r=y_1\\ \theta = \cos^{-1}y_2, 2\pi-\cos^{-1}y_2 \end{cases} $
, whose Jacobian determinant is
$\det(\frac{\partial y}{\partial \omega})= \det( \begin{bmatrix} 1 & 0\\ 0 & -\sin\theta \end{bmatrix} ) =-\sin\theta $,
so that $pdf(y_1,y_2) = \sum_{\theta:y_2=\cos \theta} pdf(r,\theta) {\lvert \det(\frac{\partial y}{\partial \omega}) \rvert }^{-1}$.
This is where I'm stuck: can I express $-\sin\theta$ in terms of $y_2$? How should I separate the cases?